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<p>Consider a heat conduction problem for a straight bar of uniform cross section and homogeneous material. Let the <span class="process-math">\(x\)</span>-axis be chosen to lie along the axis of the bar, and let <span class="process-math">\(x = 0\)</span> and <span class="process-math">\(x = L\)</span> denote the ends of the bar. Suppose further that the sides of the bar are perfectly insulated so that no heat passes through them and the ends of the bar are held at fixed temperature <span class="process-math">\(0\text{.}\)</span> We also assume that the cross-sectional dimensions are so small that the temperature <span class="process-math">\(u\)</span> can be considered as constant on any given cross section. Then <span class="process-math">\(u\)</span> is a function only of the axial coordinate <span class="process-math">\(x\)</span> and the time <span class="process-math">\(t\text{.}\)</span></p>
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